Proposition 9.4.1. The underlying anima of \(\Vect (\pt )^{\simeq }\) is the disjoint union \(\bigsqcup _{n \geq 0} \bB U(n)\). Under this isomorphism, the addition map \[ \left (\bigsqcup _{k=0}^{\infty } \bB U(k)\right ) \times \left (\bigsqcup _{l=0}^{\infty } \bB U(l)\right ) \to \left (\bigsqcup _{m=0}^{\infty } \bB U(m)\right ) \] is given on the \((k,l)\)-component by the map \(\bB U(k) \times \bB U(l) \to \bB U(k + l)\) induced by the block-sum inclusion \[ U(k) \times U(l) \hookrightarrow U(k + l), \qquad (A,B) \mapsto \begin {pmatrix} A & 0 \\ 0 & B \end {pmatrix}. \] The multiplication map is given on the \((k,l)\)-component by the map \(\bB U(k)\times \bB U(l)\to \bB U(kl)\) induced by the tensor-product representation \[ U(k)\times U(l)\longrightarrow U(kl), \qquad (A,B)\longmapsto A\otimes B. \] In particular, the induced semiring structure on \(\pi _0 \Vect (\pt ) \cong \N \) is the usual one.
Proof. By Definition 19.7.8, Proposition 19.7.9, the rank \(n\) component of \(\Vect (\pt )^{\simeq }\) is the geometric realization of the simplicial groupoid whose value at \([q]\) is the full subgroupoid \[ \left (\Vect ^{\disc }(\abs {\Delta ^q})^{\simeq }\right )_{\rk =n} \] spanned by the rank \(n\) bundles. Every rank \(n\) vector bundle over \(\abs {\Delta ^q}\) is trivial. Consequently, the full subgroupoid on the standard trivial bundle \(\abs {\Delta ^q}\times \C ^n\) is an equivalent one-object groupoid. Its automorphism group is \[ \Hom _{\Top }(\abs {\Delta ^q},\GL _n(\C )), \] and these groups assemble into the singular simplicial group of \(\GL _n(\C )\). Geometric realization preserves finite products by Lemma 21.6.12, so it commutes with the formation of classifying animae of simplicial groups. Since the realization of the singular simplicial group is the underlying group anima of \(\GL _n(\C )\), the realization of these one-object groupoids is \(\bB \GL _n(\C )\). Taking the disjoint union over all ranks therefore gives \[ \Vect (\pt )^{\simeq } \simeq \bigsqcup _{n \geq 0} \bB \GL _n(\C ). \] The inclusions \(U(n) \hookrightarrow \GL _n(\C )\) are homotopy equivalences by polar decomposition and commute with block sums and tensor products. They therefore identify \(\bB \GL _n(\C )\) with \(\bB U(n)\) compatibly with both operations.
Indeed, direct sum sends \((\C ^k,\C ^l)\) to \(\C ^{k+l}\) and induces the block-sum inclusion on automorphism groups. Similarly, tensor product sends \((\C ^k,\C ^l)\) to \(\C ^k\otimes \C ^l\cong \C ^{kl}\) and induces the tensor-product representation on automorphism groups. Passing to classifying animae gives the stated formulas. □
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